Determinantal point process models on the sphere

Transcription

1 Determinanta point process modes on the sphere Jesper Møer 1, Morten Niesen 1, Emiio Porcu 2 and Ege Rubak 1 1 Department of Mathematica Sciences, Aaborg University, Denmark jm@math.aau.dk. mniesen@math.aau.dk, rubak@math.aau.dk 2 Department of Mathematics, University Federico Santa Maria, Chie emiio.porcu@uv.c arxiv: v1 [stat.me] 13 Ju 2016 Abstract We consider determinanta point processes on the d-dimensiona unit sphere S d. These are finite point processes exhibiting repusiveness and with moment properties determined by a certain determinant whose entries are specified by a so-caed kerne which we assume is a compex covariance function defined on S d S d. We review the appeaing properties of such processes, incuding their specific moment properties, density expressions and simuation procedures. Particuary, we characterize and construct isotropic DPPs modes on S d, where it becomes essentia to specify the eigenvaues and eigenfunctions in a spectra representation for the kerne, and we figure out how repusive isotropic DPPs can be. Moreover, we discuss the shortcomings of adapting existing modes for isotropic covariance functions and consider strategies for deveoping new modes, incuding a usefu spectra approach. Keywords: isotropic covariance function; joint intensities; quantifying repusiveness; Schoenberg representation; spatia point process density; spectra representation. 1 Introduction Determinanta point processes (DPPs) are modes for repusiveness (inhibition or reguarity) between points in space, where the two most studied cases of space is a finite set or the d-dimensiona Eucidean space R d, though DPPs can be defined on fairy genera state spaces, cf. [15] and the references therein. DPPs are of interest because of their appications in mathematica physics, combinatorics, random-matrix theory, machine earning, and spatia statistics (see [19] and the references therein). For DPPs on R d, rather fexibe parametric modes can be constructed and ikeihood and moment based inference procedures appy, see [18, 19]. This paper concerns modes for DPPs defined on the d-dimensiona unit sphere S d = {x R d+1 : x = 1}, where d {1, 2,...} and x denotes the usua Eucidean distance, and where d = 1, 2 are the practicay most reevant cases. To the best of our knowedge, DPPs on S d are argey unexpored in the iterature, at east from a statistics perspective. 1

2 + Figure 1: Northern hemisphere of three spherica point patterns projected to the unit disc with an equa-area azimutha projection. Each pattern is a simuated reaization of a determinanta point process on the sphere with mean number of points 400. Left: Compete spatia randomness (Poisson process). Midde: Mutiquadric mode with τ = 10 and δ = 0.74 (see Section 4.3.2). Right: Most repusive DPP (see Section 4.2). Section 2 provides the precise definition of a DPP on S d. Briefy, a DPP on S d is a random finite subset X S d whose distribution is specified by a function C : S d S d C caed the kerne, where C denotes the compex pane, and where C determines the moment properties: the nth order joint intensity for the DPP at pairwise distinct points x 1,..., x n S d agrees with the determinant of the n n matrix with (i, j)th entry C(x i, x j ). As in most other work on DPPs, we restrict attention to the case where the kerne is a compex covariance function. We aow the kerne to be compex, since this becomes convenient when considering simuation of DPPs, but the kerne has to be rea if it is isotropic (as argued in Section 4). As discussed in Section 2, C being a covariance function impies repusiveness, and a Poisson process is an extreme case of a DPP. The eft pane in Figure 1 shows a reaization of a Poisson process whie the right pane shows a most repusive DPP which is another extreme case of a DPP studied in Section 4.2. The midde pane shows a reaization of a so-caed mutiquadric DPP where the degree of repusiveness is between these two extreme cases (see Section 4.3.2). Section 3 discusses existence conditions for DPPs and summarizes some of their appeaing properties: their moment properties and density expressions are known, and they can easiy and quicky be simuated. These resuts depend heaviy on a spectra representation of the kerne based on Mercer s theorem. Thus finding the eigenvaues and eigenfunctions becomes a centra issue, and in contrast to DPPs on R d where approximations have to be used (see [18, 19]), we are abe to hande isotropic DPPs modes on S d, i.e., when the kerne is assumed to be isotropic. Section 4, which is our main section, therefore focuses on characterizing and constructing DPPs modes on S d with an isotropic kerne C(x, y) = C 0 (s) where s = s(x, y) is the geodesic (or orthodromic or great-circe) distance and where C 0 is continuous and ensures that C becomes a covariance function. For recent efforts on covariance functions depending on the great circe distance, see [10, 3, 22]. As detaied in Section 4.1, C 0 has a Schoenberg representation, i.e., it is a countabe inear combination of Gegenbauer poynomias (cosine functions if d = 1; Legendre poynomias if d = 2) where the coefficients are nonnegative and summabe. 2

3 We denote the sum of these coefficients by η, which turns out to be the expected number of points in the DPP. In particuar we reate the Schoenberg representation to the Mercer spectra representation from Section 3, where the eigenfunctions turn out to be compex spherica harmonic functions. Thereby we can construct a number of tractabe and fexibe parametric modes for isotropic DPPs, by either specifying the kerne directy or by using a spectra approach. Furthermore, we notice the trade-off between the degree of repusiveness and how arge η can be, and we figure out what the most repusive isotropic DPPs are. We aso discuss the shortcomings of adapting existing modes for isotropic covariance functions (as reviewed in [10]) when they are used as kernes for DPPs on S d. Section 5 contains our concuding remarks, incuding future work on anisotropic DPPs on S d. 2 Preiminaries Section 2.1 defines and discusses what is meant by a DPP on S d in terms of joint intensities, Section 2.2 specifies certain reguarity conditions, and Section 2.3 discusses why there is repusiveness in a DPP. 2.1 Definition of a DPP on the sphere We need to reca a few concepts and to introduce some notation. For d = 1, 2,..., et ν d be the d-dimensiona surface measure on S d R d+1, see e.g. [6, Chapter 1]. This can be defined recursivey: For d = 1 and x = (x 1, x = (cos θ, sin θ) with 0 θ < 2π, dν 1 (x) = dθ is the usua Lebesgue measure on [0, 2π). For d 2 and x = (y sin ϑ, cos ϑ) with y S d 1 and ϑ [0, π], dν d (x) = sin d 1 ϑ dν d 1 (y) dϑ. In particuar, if d = 2 and x = (x 1, x 2, x 3 ) = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ) where ϑ [0, π] is the poar atitude and ϕ [0, 2π) is the poar ongitude, dν 2 (x) = sin ϑ dϕ dϑ. Note that S d has surface measure σ d = ν d (S d ) = 2π(d+1)/2 (σ Γ((d+1)/2) 1 = 2π, σ 2 = 4π). Consider a finite point process on S d with no mutipe points; we can view this as a random finite set X S d. For n = 1, 2,..., suppose X has nth order joint intensity ρ (n) with respect to the product measure ν (n) d = ν d ν d (n times), that is, for any Bore function h : (S d ) n [0, ), E x 1,...,x n X h(x 1,..., x n ) = h(x 1,..., x n )ρ (n) (x 1,..., x n ) dν (n) d (x 1,..., x n ), (2.1) where the expectation is with respect to the distribution of X and over the summation sign means that the sum is over a x 1 X,..., x n X such that x 1,..., x n are pairwise different, so uness X contains at east n points the sum is zero. In particuar, ρ(x) = ρ (1) (x) is the intensity function (with respect to ν d ). Intuitivey, if x 1,..., x n are pairwise distinct points on S d, then ρ (n) (x 1,..., x n ) dν (n) d (x 1,..., x n ) 3

4 is the probabiity that X has a point in each of n infinitesimay sma regions on S d around x 1,..., x n and of sizes dν d (x 1 ),..., dν d (x n ), respectivey. Note that ρ (n) is uniquey determined except on a ν (n) d -nuset. Definition 2.1. Let C : S d S d C be a mapping and X S d be a finite point process. We say that X is a determinanta point process (DPP) on S d with kerne C and write X DPP d (C) if for a n = 1, 2,... and x 1,..., x n S d, X has nth order joint intensity ρ (n) (x 1,..., x n ) = det (C(x i, x j ) i,j=1,...,n ), (2.2) where det (C(x i, x j ) i,j=1,...,n ) is the determinant of the n n matrix with (i, j)th entry C(x i, x j ). Comments to Definition 2.1: (a) If X DPP d (C), its intensity function is ρ(x) = C(x, x), x S d, and the trace η = C(x, x) dν d (x) (2.3) is the expected number of points in X. (b) A Poisson process on S d with a ν d -integrabe intensity function ρ is a DPP where the kerne on the diagona agrees with ρ and outside the diagona is zero. Another simpe case is the restriction of the kerne of the Ginibre point process defined on the compex pane to S 1, i.e., when C(x 1, x = ρ exp [exp {i (θ 1 θ }], x k = exp (iθ k ) S 1, k = 1, 2. (2.4) (The Ginibre point process defined on the compex pane is a famous exampe of a DPP and it reates to random matrix theory, see e.g. [8, 15]; it is ony considered in this paper for iustrative purposes.) (c) In accordance with our intuition, condition (2.2) impies that ρ (n+1) (x 0,..., x n ) > 0 ρ (n) (x 1,..., x n ) > 0 for any pairwise distinct points x 0,..., x n S d with n 1. Condition (2.2) aso impies that C must be positive semi-definite, since ρ (n) 0. In particuar, by (2.2), C is (stricty) positive definite if and ony if ρ (n) (x 1,..., x n ) > 0 for n = 1, 2,... and pairwise distinct points x 1,..., x n S d. (2.5) The impication of the kerne being positive definite wi be discussed severa paces further on. 4

5 2.2 Reguarity conditions for the kerne Henceforth, as in most other pubications on DPPs (defined on R d or some other state space), we assume that C in Definition 2.1 is a compex covariance function, i.e., C is positive semi-definite and Hermitian, is of finite trace cass, i.e., η <, cf. (2.3), and C L 2 (S d S d, ν (2) d ), the space of ν(2) square integrabe compex functions. These reguarity conditions become essentia when we ater work with the spectra representation for C and discuss various properties of DPPs in Sections 3 4. Note that if C is continuous, then η < and C L 2 (S d S d, ν (2) d ). For instance, the reguarity conditions are satisfied for the Ginibre DPP with kerne (2.4). 2.3 Repusiveness Since C is a covariance function, condition (2.2) impies that d ρ (n) (x 1,..., x n ) ρ(x 1 ) ρ(x n ), (2.6) with equaity ony if X is a Poisson process with intensity function ρ. Therefore, since a Poisson process is the case of no spatia interaction, a non-poissonian DPP is repusive. For x, y S d, et C(x, y) R(x, y) = C(x, x)c(y, y) be the correation function corresponding to C when ρ(x)ρ(y) > 0, and define the pair correation function for X by g(x, y) = { ρ (2) (x,y) ρ(x)ρ(y) = 1 R(x, y) 2 if ρ(x)ρ(y) > 0. 0 otherwise. (2.7) (This terminoogy for g may be confusing, but it is adapted from physics and is commony used by spatia statisticians.) Note that g(x, x) = 0 and g(x, y) 1 for a x y, with equaity ony if X is a Poisson process, again showing that a DPP is repusive. 3 Existence, simuation, and density expressions Section 3.1 recas the Mercer (or spectra) representation for a compex covariance function. This is used in Section 3.2 to describe the existence condition and some basic probabiistic properties of X DPP d (C), incuding a density expression for X which invoves a certain kerne C. Finay, Section 3.3 notices an aternative way of specifying a DPP, namey in terms of the kerne C. 5

6 3.1 Mercer representation We need to reca the spectra representation for a compex covariance function K : S d S d C which coud be the kerne C of a DPP or the above-mentioned kerne C. Assume that K is of finite trace cass and is square integrabe, cf. Section 2.2. Then, by Mercer s theorem (see e.g. [25, Section 98]), ignoring a ν (2) d -nuset, we can assume that K has spectra representation with K(x, y) = α n Y n (x)y n (y), x, y S d, (3.1) n=1 absoute convergence of the series; Y 1, Y 2,... being eigenfunctions which form an orthonorma basis for L 2 (S d, ν d ), the space of ν d square integrabe compex functions; the set of eigenvaues spec(k) = {α 1, α 2,...} being unique, where each nonzero α n is positive and has finite mutipicity, and the ony possibe accumuation point of the eigenvaues is 0; see e.g. [15, Lemma 4.2.2]. If in addition K is continuous, then (3.1) converges uniformy and Y n is continuous if α n 0. We refer to (3.1) as the Mercer representation of K and ca the eigenvaues for the Mercer coefficients. Note that spec(k) is the spectrum of K. When X DPP d (C), we denote the Mercer coefficients of C by λ 1, λ 2,.... By (2.3) and (3.1), the mean number of points in X is then η = λ n. (3.2) n=1 For exampe, for the Ginibre DPP with kerne (2.4), C(x 1, x = ρ n=0 exp {i n(θ 1 θ } n!. (3.3) As the Fourier functions exp(inθ), n = 0, 1,..., are orthogona, the Mercer coefficients become 2πρ/n!, n = 0, 1, Resuts Theorem 3.2 beow summarizes some fundamenta probabiistic properties for a DPP. First we need a definition, noticing that in the Mercer representation (3.1), if spec(k) {0, 1}, then K is a projection, since K(x, z)k(z, y) dν d (z) = K(x, y). Definition 3.1. If X DPP d (C) and spec(c) {0, 1}, then X is caed a determinanta projection point process. 6

7 Theorem 3.2. Let X DPP d (C) where C is a compex covariance function of finite trace cass and C L 2 (S d S d, ν (2) d ). (a) Existence of DPP d (C) is equivaent to that and it is then unique. spec(c) [0, 1] (3.4) (b) Suppose spec(c) [0, 1] and consider the Mercer representation C(x, y) = λ n Y n (x)y n (y), x, y S d, (3.5) n=1 and et B 1, B 2,... be independent Bernoui variabes with means λ 1, λ 2,.... Conditiona on B 1, B 2,..., et Y DPP d (E) be the determinanta projection point process with kerne E(x, y) = B n Y n (x)y n (y), x, y S d. n=1 Then X is distributed as Y (unconditionay on B 1, B 2,...). (c) Suppose spec(c) {0, 1}. Then the number of points in X is constant and equa to η = C(x, x) dν d (x) = #{n : λ n = 1}, and its density with respect to ν (η) d is f η ({x 1,..., x n }) = 1 η! det (C(x i, x j ) i,j=1,...,η ), {x 1,..., x η } S d. (3.6) (d) Suppose spec(c) [0, 1). Let C : S d S d C be the compex covariance function given by the Mercer representation sharing the same eigenfunctions as C in (3.5) but with Mercer coefficients Define λ n = λ n 1 λ n, n = 1, 2,... (3.7) D = og(1 + λ n ). n=1 Then DPP d (C) is absoutey continuous with respect to the Poisson process on S d with intensity measure ν d and has density f({x 1,..., x n }) = exp (σ d D) det ( C(xi, x j ) i,j=1,...,n ), (3.8) for any finite point configuration {x 1,..., x n } S d (n = 0, 1,...). (e) Suppose spec(c) [0, 1) and C is (stricty) positive definite. Then any finite subset of S d is a feasibe reaization of X, i.e., f({x 1,..., x n }) > 0 for a n = 0, 1,... and pairwise distinct points x 1,..., x n S d. 7

8 Comments to Theorem 3.2: (a) This foows from [15, Lemma and Theorem 4.5.5]. For exampe, for the Ginibre DPP on S d, it foows from (3.3) that η 1, so this process is of very imited interest in practice. We sha ater discuss in more detai the impication of the condition (3.4) for how arge the intensity and how repusive a DPP can be. Note that (3.4) and C being of finite trace cass impy that C L 2 (S d S d, ν (2) d ). (b) This fundamenta resut is due to [14, Theorem 7] (see aso [15, Theorem 4.5.3]). It is used for simuating a reaization of X in a quick and exact way: Generate first the finitey many non-zero Bernoui variabes and second in a sequentia way each of the n=1 B n points in Y, where a joint density simiar to (3.6) is used to specify the conditiona distribution of a point in Y given the Bernoui variabes and the previousy generated points in Y. In [18, 19] the detais for simuating a DPP defined on a d-dimensiona compact subset of R d are given, and with a change to spherica coordinates this procedure can immediatey be modified to appy for a DPP on S d (see Appendix A for an important technica detai which differs from R d ). (c) This resut for a determinanta projection point process is in ine with (b). (d) For a proof of (3.8), see e.g. [27, Theorem 1.5]. If n = 0 then we consider the empty point configuration. Thus exp( D) is the probabiity that X =. Moreover, we have the foowing properties: f is hereditary, i.e., for n = 1, 2,... and pairwise distinct points x 0,..., x n S d, f({x 0,..., x n }) > 0 f({x 1,..., x n }) > 0. (3.9) In other words, any subset of a feasibe reaization of X is aso feasibe. If Z denotes a unit rate Poisson process on S d, then ρ (n) (x 1,..., x n ) = Ef(Z {x 1,..., x n }) (3.10) for any n = 1, 2,... and pairwise distinct points x 1,..., x n S d. C is of finite trace cass and C L 2 (S d S d, ν (2) d ). There is a one-to-one correspondence between C and C, where λ n = (e) This foows by combining (2.5), (3.9), and (3.10). 3.3 Defining a DPP by its density λ n 1 + λ n, n = 1, 2,... (3.11) Aternativey, instead of starting by specifying the kerne C of a DPP on S d, if spec(c) [0, 1), the DPP may be specified in terms of C from the density expression (3.8) by expoiting the one-to-one correspondence between C and C: First, we assume 8

9 that C : S d S d C is a covariance function of finite trace cass and C L 2 (S d S d, ν (2) d ) (this is ensured if e.g. C is continuous). Second, we construct C from the Mercer representation of C, recaing that C and C share the same eigenfunctions and that the Mercer coefficients for C are given in terms of those for C by (3.11). Indeed then spec(c) [0, 1) and λ n <, and so DPP d (C) is we defined. 4 Isotropic DPP modes Throughout this section we assume that X DPP d (C) where C is a continuous isotropic covariance function with spec(c) [0, 1], cf. Theorem 3.2(a). Here isotropy means that C is invariant under the action of the orthogona group O(d + 1) on S d. In other words, C(x, y) = C 0 (s), where s = s(x, y) = arccos(x y), x, y S d, is the geodesic (or orthodromic or great-circe) distance and denotes the usua inner product on R d+1. Thus C being Hermitian means that C 0 is a rea mapping: since C(x, y) = C 0 (s) = C(y, x) and C(x, y) = C(y, x), we see that C(y, x) = C(y, x) is rea. Therefore, C 0 is assumed to be a continuous mapping defined on [0, π] such that C becomes positive semi-definite. Moreover, we foow [7] in caing C 0 : [0, π] R the radia part of C, and with itte abuse of notation we write X DPP d (C 0 ). Note that some specia cases are excuded: For a Poisson process with constant intensity, C is isotropic but not continuous. For the Ginibre DPP, the kerne (2.4) is a continuous covariance function, but since the kerne is not rea it is not isotropic (the kerne is ony invariant under rotations about the origin in the compex pane). Obviousy, X is invariant in distribution under the action of O(d + 1) on S d. In particuar, any point in X is uniformy distributed on S d. Further, the intensity ρ = C 0 (0) is constant and equa to the maxima vaue of C 0, whie η = σ d C 0 (0) is the expected number of points in X. Furthermore, assuming C 0 (0) > 0 (otherwise X = ), the pair correation function is isotropic and given by where g(x, y) = g 0 (s) = 1 R 0 (s) 2, (4.1) R 0 (s) = C 0 (s)/c 0 (0) is (the radia part of) the correation function associated to C. Note that g 0 (0) = 0. For many exampes of isotropic kernes for DPPs (incuding those discussed ater in this paper), g 0 wi be a non-decreasing function (one exception is the most repusive DPP given in Proposition 4.4 beow). In what foows, since we have two kinds of specifications for a DPP, namey in terms of C or C (where in the atter case spec(c) [0, 1), cf. Section 3.3), et us 9

10 just consider a continuous isotropic covariance function K : S d S d R. Our aim is to construct modes for its radia part K 0 so that we can cacuate the Mercer coefficients for K and the corresponding eigenfunctions and thereby can use the resuts in Theorem 3.2. As we sha see, the case d = 1 can be treated by basic Fourier cacuus, whie the case d 2 is more compicated and invoves surface spherica harmonic functions and so-caed Schoenberg representations. In the seque, without oss of generaity, we assume K 0 (0) > 0 and consider the normaized function K 0 (s)/k 0 (0), i.e., the radia part of the corresponding correation function. Section 4.1 characterizes such functions so that in Section 4.2 we can quantify the degree of repusiveness in an isotropic DPP and in Section 4.3 we can construct exampes of parametric modes. 4.1 Characterization of isotropic covariance functions on the sphere Gneiting [10] provided a detaied study of continuous isotropic correation functions on the sphere, with a view to (Gaussian) random fieds defined on S d. This section summarizes the resuts in [10] needed in this paper and compement with resuts reevant for DPPs. For d = 1, 2,..., et Ψ d be the cass of continuous functions ψ : [0, π] R such that ψ(0) = 1 and the function R d (x, y) = ψ(s), x, y S d, (4.2) is positive semi-definite, where the notation stresses that R d depends on d (i.e., R d is a continuous isotropic correation function defined on S d S d ). The casses Ψ d and Ψ = d=1 Ψ d are convex, cosed under products, and cosed under imits if the imit is continuous, cf. [26]. Let Ψ + d be the subcass of those functions ψ Ψ d which are (stricty) positive definite, and set Ψ + = d=1 Ψ+ d, Ψ d = Ψ d \ Ψ + d, and Ψ = d=1 Ψ d. By [10, Coroary 1], these casses are stricty decreasing: Ψ 1 Ψ 2 Ψ, Ψ + 1 Ψ + 2 Ψ +, Ψ 1 Ψ 2 Ψ, and Ψ = Ψ + Ψ, where the union is disjoint. The foowing Theorem 4.1 characterizes the cass Ψ d in terms of Gegenbauer poynomias and so-caed d-schoenberg coefficients (this terminoogy is adapted from [7]). It aso estabishes the connection to the Mercer representation of a continuous isotropic correation function. Reca that the Gegenbauer poynomia C (λ) : [ 1, 1] R of degree = 0, 1,... is defined for λ > 0 by the expansion 1 (1 + r 2 2r cos s) = r C (λ) λ (cos s), 1 < r < 1, 0 s π. (4.3) We foow [26] in defining =0 C (0) (cos s) = cos(s), 0 s π. 10

11 We have C (0) (1) = 1 and C ( d 1 (1) = ( ) +d 2 for d = 2, 3,.... Further, the Legendre poynomia of degree = 0, 1,... is by Rodrigues formua given by P (x) = 1 d 2! dx {(x2 1) }, 1 < x < 1, and for m = 0,...,, the associated Legendre functions P (m) by and and P ( m) P (m) (x) = ( 1) m ( 1 x 2) m/2 d m dx m P (x), 1 x 1, P ( m) m ( m)! = ( 1) ( + m)! P (m). are given Note that C ( 1 = P. Furthermore, in Theorem 4.1(b), Y,k,d is a compex spherica harmonic function and K,d is an index set such that the functions Y,k,d for k K,d and = 0, 1,... are forming an orthonorma basis for L 2 (S d, ν d ). Compex spherica harmonic functions are constructed in e.g. [12, Eq. (2.5)]), but since their genera expression is rather compicated, we have chosen ony to specify these in Theorem 4.1(c) (d) for the practicay most reevant cases d = 1, 2. Finay, etting m,d = #K,d, then m 0,1 = 1, m,1 = 2, = 1, 2,..., and m,d = 2 + d 1 d 1 ( + d 2 ), = 0, 1,..., d = 2, 3,..., (4.4) i.e., m,2 = for = 0, 1,.... Theorem 4.1. We have: (a) ψ Ψ d if and ony if ψ is of the form ψ(s) = C ( d 1 (cos s) β,d, 0 s π, (4.5) C ( d 1 (1) =0 where the d-schoenberg sequence β 0,d, β 1,d,... is a probabiity mass function. Then, for d = 1, ψ Ψ + 1 if and ony if for any two integers 0 < n, there exists an integer k 0 such that β +kn,1 > 0. Whie, for d 2, ψ Ψ + d if and ony if the subsets of d-schoenberg coefficients β,d > 0 with an even respective odd index are infinite. (b) For the correation function R d in (4.2) with ψ Ψ d given by (4.5), the Mercer representation is R d (x, y) = =0 α,d k K,d Y,k,d (x)y,k,d (y), (4.6) 11

12 where the Mercer coefficient α,d is an eigenvaue of mutipicity m,d and it is reated to the d-schoenberg coefficient β,d by α,d = σ d β,d m,d, = 0, 1,... (4.7) (c) Suppose d = 1. Then the Schoenberg representation (4.5) becomes Conversey, β 0,1 = 1 π π 0 ψ(s) ds, ψ(s) = β,1 cos(s), 0 s π. (4.8) =0 β,1 = 2 π π 0 cos(s)ψ(s) ds, = 1, 2,... (4.9) Moreover, for R 1 given by the Mercer representation (4.6), we have K 0,1 = {0} and K,1 = {±1} for > 0, and the eigenfunctions are the Fourier basis functions for L 2 (S 1, ν 1 ): Y,k,1 (θ) = exp(ikθ) 2π, 0 θ < 2π, = 0, 1,..., k K,1. (d) Suppose d = 2. Then the Schoenberg representation (4.5) becomes ψ(s) = β,2 P (cos s), 0 s π. =0 Moreover, for R 2 given by the Mercer representation (4.6), K,2 = {,..., } and the eigenfunctions are the surface spherica harmonic functions given by ( k)! Y,k,2 (ϑ, ϕ) = 4π ( + k)! P (k) (cos ϑ) e ikϕ, (4.10) Comments to Theorem 4.1: (ϑ, ϕ) [0, π] [0, 2π), = 0, 1,..., k K,2. (a) Expression (4.5) is a cassica characterization resut due to Schoenberg [26]. For the other resuts in (a), see [10, Theorem 1]. (b) For d = 1, (4.6) is straightforwardy verified using basic Fourier cacuus. For d 2, (4.6) foows from (4.5), where C ((d 1)/2) (1) = ( ) +d 2, and from the genera addition formua for spherica harmonics (see e.g. [6, p. 10]): Y,k,d (x)y,k,d (y) = d 1 σ d d 1 k K,d C ( d 1 (cos s). (4.11) When R d in (4.6) is the correation function R 0 for the kerne C of the isotropic DPP X with intensity ρ, note that λ,k,d = λ,d = η m,d β,d, k K,d, = 0, 1,..., (4.12) 12

13 are the Mercer coefficients for C. Hence the range for the intensity is m,d 0 < ρ ρ max,d, ρ max,d = inf, (4.13) : β,d >0 σ d β,d where ρ max,d is finite and as indicated in the notation may depend on the dimension d. In the specia case where ψ(s) is nonnegative, we prove in Appendix B that the infimum in (4.13) is attained at = 0 and consequenty ρ max,d = m 0,d σ d β 0,d. (4.14) Note that the condition spec(c) [0, 1) is equivaent to ρ < ρ max,d, and the kerne C used in the density expression (3.8) is then as expected isotropic, with Mercer coefficients λ,k,d = λ,d = ηβ,d m,d ηβ,d, k K,d, = 0, 1,.... This foows by combining (3.7), (4.5), (4.7) and (4.12). (c) This foows straightforwardy from basic Fourier cacuus. (d) These resuts foow from (4.5) (4.7). (The reader mainy interested in the proof for case d = 2 may consut [20, Proposition 3.29] for the fact that the surface spherica harmonics given by (4.10) constitute an orthonorma basis for L 2 (S 2, ν, and then use (4.11) where C ((d 1)/2) = P for d = 2.) For d = 1, the inversion resut (4.9) easiy appies in many cases. When d 2, [10, Coroary 2] (based on [26]) specifies the d-schoenberg coefficients: β,d = 2 + d 1 (Γ( d 1 π )2 2 3 d π Γ(d 1) 0 C ( d 1 (cos s) sin d 1 (s)ψ(s) ds, = 0, 1,... (4.15) In genera we find it hard to use this resut, whie it is much easier first to find the so-caed Schoenberg coefficients β given in the foowing theorem and second to expoit their connection to the d-schoenberg coefficients (stated in (4.17) beow). We need some further notation. For non-negative integers n and, define for d = 1, γ (1) n, = ( ) ( ) 2 2 δ n,0 δ (mod 2),0 n, 2 where δ ij is the Kronecker deta, and define for d = 2, 3,..., γ (d) n, Theorem 4.2. We have: d 1 (2n + d 1)(!)Γ( = 2 +1 {( n 2 ) 2 +n+d+1 )!}Γ( ) 2 ( n + d 2 n ). 13

14 (a) ψ Ψ if and ony if ψ is of the form ψ(s) = β cos s, 0 s π, (4.16) =0 where the Schoenberg sequence β 0, β 1,... is a probabiity mass function. Moreover, ψ Ψ + if and ony if the subsets of Schoenberg coefficients β > 0 with an even respective odd index are infinite. (b) For ψ Ψ and d = 1, 2,..., the d-schoenberg sequence is given in terms of the Schoenberg coefficients by β n,d = Comments to Theorem 4.2: =n n 0 (mod 2) β γ (d) n,, n = 0, 1,..., (4.17) (a) Expression (4.16) is a cassica characterization resut due to Schoenberg [26], whie we refer to [10, Theorem 1] for the remaining resuts. It is usefu to rewrite (4.16) in terms of a probabiity generating function ϕ(x) = x β, 1 x 1, =0 so that ψ(s) = ϕ(cos s). Exampes are given in Section 4.3. (b) The reation (4.17) is verified in Appendix C. Given the Schoenberg coefficients, (4.17) can be used to cacuate the d- Schoenberg coefficients either exacty or approximatey by truncating the sums in (4.17). If there are ony finitey many non-zero Schoenberg coefficients in (4.16), then there is ony finitey many non-zero d-schoenberg coefficients and the sums in (4.17) are finite. Exampes are given in Section Quantifying repusiveness Consider again X DPP d (C 0 ) where C 0 = ρr 0, ρ > 0 is the intensity, and R 0 is the correation function. For distinct points x, y S d, reca that ρ 2 g 0 (s(x, y)) dν d (x) dν d (y) is approximatey the probabiity for X having a point in each of infinitesimay sma regions on S d around x and y of sizes dν d (x) and dν d (y), respectivey. Therefore, when seeing if a DPP is more repusive than another by comparing their pair correation functions, we need to fix the intensity. Naturay, we wi say that X (1) DPP d (C (1) 0 ) is at east as repusive than X (2) DPP d (C (2) 0 ) if they share the same intensity and their pair correation functions satisfy g (1) 0 g (2) 0 (using an obvious notation). However, as pointed out in [18, 19] such a simpe comparison is not aways possibe. 14

15 Instead, foowing [18] (see aso [4]), for an arbitrary chosen point x S d, we quantify goba repusiveness of X by I(g 0 ) = 1 [1 g 0 {s(x, y)}] dν d (y) = 1 R 0 {s(x, y)} 2 dν d (y). σ d S σ d d S d Ceary, I(g 0 ) does not depend on the choice of x, and 0 I(g 0 ) 1, where the ower bound is attained for a Poisson process with constant intensity. Furthermore, assuming R 0 (s) is twice differentiabe from the right at s = 0, we quantify oca repusiveness of X by the sope g 0(0) = 2R 0 (0)R 0(0) = 2R 0(0) of the tangent ine of the pair correation function at s = 0 and by its curvature c(g 0 ) = g 0(0) R = 2 0(0) 2 + R 0(0) {1 + g 0 2 3/2 (0)} {1 + 4R 0(0) 2 }. 3/2 For many modes we have R 0(0) = 0, and so g 0(0) = 0 and c(g 0 ) = g 0(0). In some cases, the derivative of R 0 (s) has a singuarity at s = 0 (exampes are given in Section 4.3.5); then we define g 0(0) =. Definition 4.3. Suppose X (1) DPP d (C (1) 0 ) and X (2) DPP d (C (2) 0 ) share the 0, respectivey. We say that X (1) is at east as gobay repusive than X (2) if I(g (1) 0 ) I(g (2) 0 ). We say that X (1) is ocay more repusive than X (2) if either g (1) 0 (s) and g (2) 0 (s) are differentiabe at s = 0 with g (1) 0 (0) < g (2) 0 (0) or if g (1) 0 (s) and g (2) 0 (s) are twice same intensity ρ > 0 and have pair correation functions g (1) 0 and g (2) differentiabe at s = 0 with g (1) 0 (0) = g (2) 0 (0) and c(g (1) 0 ) < c(g (2) We think of the homogeneous Poisson process as the east gobay and ocay repusive DPP (for a given intensity), since its pair correation function satisfies g 0 (0) = 0 and g 0 (s) = 1 for 0 s π (this wi be a imiting case in our exampes to be discussed in Section 4.3). In what foows, we determine the most gobay and ocay repusive DPPs. For η = σ d ρ > 0, et X (η) DPP d (C (η) 0 ) where C (η) 0 has Mercer coefficient λ (η),d (of mutipicity m,d ) given by λ (η),d = 1 if < n, λ(η) =0 0 ). n,d = 1 ( n 1 ) η m,d, λ (η),d = 0 if > n, (4.18) m n,d where n 0 is the integer such that n 1 =0 m,d < η n =0 m,d, and where we set = 0. That is, 1 =0 C (η) 0 (s) = 1 n 1 σ d =0 C ( d 1 m,d (cos s) + 1 n 1 (η C ( d 1 (1) σ d =0 d 1 ) ( C m,d n (cos s). (4.19) C ( d 1 n (1) If η = n =0 m,d, then X (η) is a determinanta projection point process consisting of η points, cf. Theorem 3.2(b). If η < n =0 m,d, then X (η) is approximatey a determinanta projection point process and the number of points in X (η) is random with vaues in { n 1 =0 m,d, 1+ n 1 =0 m,d,..., n =0 m,d}. The foowing proposition is verified in Appendix D. 15

16 Theorem 4.4. For a fixed vaue of the intensity ρ > 0, we have: (a) I(g 0 ) satisfies (b) If ηi(g 0 ) = 1 1 η m,d λ,d (1 λ,d ), (4.20) =0 and so X (η) is a gobay most repusive isotropic DPP. then g 0(0) = 0 and 2 β,d <, (4.21) =1 c(g 0 ) = g 0(0) = 2 d ( + d 1)β,d. (4.22) (c) X (η) is the unique ocay most repusive DPP among a isotropic DPPs satisfying (4.21). Comments to Theorem 4.4: (a) It foows from (4.20) that there may not be a unique gobay most repusive isotropic DPP, however, X (η) appears to be the most natura one. For instance, if η = n =0 m,d, there may exist another gobay most repusive determinanta projection point process with the non-zero Mercer coefficients specified by another finite index set L {0, 1,...} than {0,..., n}. In particuar, for d = 1 and n > 0, there are infinitey many such index sets. By (4.20), ηi(g 0 ) 1, =1 where the equaity is obtained for g 0 = g (η) 0 when η = n =0 m,d. This inverse reationship between η and I(g 0 ) shows a trade-off between intensity and the degree of repusiveness in a DPP. (b) The variance condition (4.21) is sufficient to ensure twice differentiabiity from the right at 0 of R 0. The condition is vioated in the case of the exponentia covariance function (the Matérn covariance function with ν = 1/2 and studied in Section 4.3.5). (c) For simpicity, suppose that η = n =0 m,d. Then X (η) is a determinanta projection point process consisting of η points and with pair correation function g (η) 0 (0) = 2 n =1 ( + d 1)m,d d n =0 m, (4.23),d cf. (4.18) and (4.22). Note that g (η) 0 (0) n 2 (here f 1 () f 2 () means that c d f 1 ()/f 2 () C d where C d c d are positive constants). 16

17 For the practica important cases d 2, we have that η = 2n + 1 is odd if d = 1, whie η = (n + 1) 2 is quadratic if d = 2, cf. (4.4). Furthermore, (4.19) simpifies to C (η) 0 (s) = 1 n cos(s) if d = 1, 2π and C (η) 0 (s) = 1 4π = n n (2 + 1)P (cos s) if d = 2. =0 Finay, a straightforward cacuation shows that (4.23) becomes and g (η) 0 (0) = 2 3 n2 + 2 n if d = 1, (4.24) 3 g (η) 0 (0) = 1 2 n2 + n if d = 2. (4.25) 4.3 Parametric modes In accordance with Theorem 4.4 we refer to X (η) as the most repusive DPP (when η is fixed). Ideay a parametric mode cass for the kerne of a DPP shoud cover a wide range of repusiveness, ranging from the most repusive DPP to the east repusive DPP (the homogeneous Poisson process). This section considers parametric modes for correation functions ψ Ψ d used to mode (i) either C 0 of the form C 0 (s) = ρψ(s), 0 < ρ ρ max,d, (4.26) where ρ max,d < is the upper bound on the intensity ensuring the existence of the DPP, cf. (4.13), noticing that ρ max,d depends on ψ; (ii) or C 0 of the form C 0 (s) = χψ(s), χ > 0, (4.27) where C 0 is the radia part for C, and so the DPP is we defined for any positive vaue of the parameter χ. In case (i) we need to determine the d-schoenberg coefficients or at east ρ max,d = ρ max,d (ψ), and the d-schoenberg coefficients wi aso be needed when working with the ikeihood, cf. (3.8). In case (ii) we can immediatey work with the ikeihood, whie we need to cacuate the d-schoenberg coefficients in order to find the intensity and the pair correation function. In both cases, if we want to simuate from the DPP, the d-schoenberg coefficients have to be cacuated. In case (ii) with fixed ψ, the og-ikeihood is simpe to hande with respect to the rea parameter ζ = n χ: if {x 1,..., x n } is an observed point pattern with n > 0 and α,d is the th Mercer coefficient for ψ, the og-ikeihood is (ζ) = nζ + n [ ] det{ψ(s(x i, x j ))} i,j=1,...,n m,d n(1 + α,d χ), 17 =0

18 cf. (3.8). Hence the score function is d(ζ) dζ = n =0 m,d α,d χ 1 + α,d χ, and the observed information is d2 (ζ) dζ 2 = =0 m,d α,d χ (1 + α,d χ) 2, which is stricty positive (and agrees with the Fisher information). Thus Newton- Raphson can be used for determining the maximum ikeihood estimate of χ Mode strategies In genera, when we start with a cosed form expression for ψ, the d-schoenberg coefficients wi be not be expressibe on cosed form. Notabe exceptions are a specia case of the mutiquadric famiy studied in Section and the spherica famiy and specia cases of the Askey and Wendand famiies (with d {1, 3}) considered in Section Instead, if ψ Ψ, the Schoenberg coefficients may be expressed in cosed form, making use of the identity in (4.17). This is possibe, for instance, for the mutiquadric mode. Other exampes can be obtained by considering any probabiity mass system with a probabiity generating function being avaiabe in cosed form. For instance, the binomia, Poisson, ogathmic famiies can be used for such a setting. However, in practice, if the sum in (4.17) is infinite, a truncation wi be needed so that approximate d-schoenberg coefficients are cacuated (these wi be smaer than the true ones, so in case (i) above the approximation of the DPP is sti a we defined DPP). For instance, this is needed in case of a Poisson distribution but not in case of a binomia distribution. When the dimension d is fixed, an aternative and as iustrated in Section often more fexibe approach is to start by modeing the Mercer coefficients for ψ. Then typicay ψ can ony be expressed as an infinite sum (its d-schoenberg representation). On the other hand, apart from the specia cases considered in Section , we have not been successfu in expressing Schoenberg coefficients on cosed form for the commony used modes for correation functions, i.e., those isted in [10, Tabe 1]: the powered exponentia, Matérn, generaized Cauchy, etc. Moreover, these commony used modes seem not very fexibe for modeing repusiveness. Section iustrates this in the case of the Matérn mode Mutiquadric covariance functions Let p (0, 1) and τ > 0 be the parameters of the negative binomia distribution ( ) τ + 1 β = p (1 p) τ, = 0, 1,... 18

19 Then the corresponding Schoenberg representation reduces to ( ) τ 1 p ψ(s) =, 0 s π, (4.28) 1 p cos s where ψ Ψ +, cf. Theorem 4.2(a). This is the same as the mutiquadric mode in [10] based on the reparametrization given by p = 2δ 1+δ 2 with δ (0, 1), since ψ(s) = (1 δ) 2τ, 0 s π. (4.29) (1 + δ 2 2δ cos s) τ For d = 2, we obtain for τ = 1 the inverse mutiquadric famiy and for τ = 3 the 2 2 Poisson spine [5]. Furthermore, for d = 2 we can sove (4.15) expicity and we have that the maxima 2-Schoenberg coefficient is { (1 δ) 2τ ( (1 + δ) 2(1 τ) (1 δ) 2(1 τ)) for τ 1 β 0,2 = 4δ(1 τ) (1 δ) 2 og ( 1+δ 2δ 1 δ ) for τ = 1 (4.30) which by (4.14) gives us ρ max,d. Suppose d 2 and τ = d 1. Then we derive directy from (4.3) and (4.29) that 2 ( ) + d 2 β,d = δ (1 δ) d 1, = 0, 1,..., (4.31) are the d-schoenberg coefficients. Consider the case (i) where C 0 = ρψ, and et η max,d = σ d ρ max,d be the maxima vaue of η = σ d ρ (the mean number of points). Then η max,d = (1 δ) 1 d, which is an increasing function of δ, with range (1, ), and λ,d = η d 1 η max,d 2 + d 1 δ, = 0, 1,... For any fixed vaue of η > 0, as δ 1, we obtain η max,d and λ,d 0, corresponding to the Poisson process with intensity ρ. On the other hand, the DPP is far from the most repusive DPP with the same vaue of η uness η is very cose to one: If η = η max,d > 1, then λ,d = d 1 ) (1 η 1 1 d, 2 + d 1 which is faster than agebraicay decaying as a function of and is a stricty decreasing function of η when > 0. Aso the variance condition (4.21) is seen to be satisfied. Hence, for η = η max,d > 1, c(g 0 ) = g (0) = 2(d 1)δ ) = 2(d 1) (η 2 1 (1 δ) 2 d 1 η d 1 which is an increasing function of η, with range (0, ), and g (0) is of order η 2 d 1. Thus we see again that the DPP is far ess repusive than the most repusive DPP 19

20 Figure 2: Pair correation functions for DPP modes with d = 2. Fu ines from eft to right correspond to mutiquadric modes with τ = 1, 2, 5, 10, 100 and δ = 0.97, 0.90, 0.82, 0.74, 0.38 chosen such that η max,2 = 400. The dotted ine corresponds to the most repusive DPP with η = 400. with the same vaue of η (recaing that g (η) 0 (0) is of order η 2/d ). To iustrate this, et d = 2 and η = η max,2 = (1 + n) 2. Then g (0) = 2 ( (n + 1) 4 (n + 1) 2), which is of order n 4, whie g ((1+n) 0 (0) = 1 2 n2 +n is of order n 2 (the case of the most repusive DPP with (1 + n) 2 points, cf. (4.25)). In concusion the inverse mutiquadric mode is not very fexibe in terms of the repusiveness it can cover. However, for other choices of τ the situation appears to be much better. Figure 2 shows the pair correation function for different vaues of τ for d = 2 when δ is chosen such that η max,2 = 400 together with the most repusive DPP with η = 400. The figure suggests that the modes become more repusive when τ with δ 0 appropriatey chosen to keep η max,2 fixed, and based on the figure we conjecture that a imiting mode exists, but we have not been abe to prove this. The simuated reaization in the midde pane of Figure 1 gives the quaitative impression that the mutiquadric mode can obtain a degree of repusiveness that amost reaches the most repusive DPP though the corresponding pair correation functions can easiy be distinguished in Figure 2. To cacuate η max,2 we simpy use (4.30) and (4.14) Spherica, Askey, and Wendand covariance functions Tabe 1 shows specia cases of Askey s truncated power function and C 2 -Wendand and C 4 -Wendand correation functions when d 3 (here, for any rea number, x + = x if x 0, and x + = 0 if x < 0). Note that a scae parameter c can be incuded, where for the C 2 and C 4 Wendand correation functions, c (0, 2π] and c defines the compact support of these correation functions, and where for the Askey s truncated power function, c > 0. Notice that these correation functions are of cass Ψ + 3, they are compacty supported for c < π, and compared to the Askey 20

21 and Wendand correation functions in [10, Tabe 1] they are the most repusive cases. Appendix E describes how the one-schoenberg coefficients isted in Tabe 1 can be derived. The one-schoenberg coefficients can then be used to obtain the 3-Schoenberg coefficients, cf. [10, Coroary 3]. Moreover, the spherica correation function ( ψ(s) = 1 + s ) ( 1 s ) 2, 0 s π, 2c c + is of cass Ψ + 3, and the proof of [10, Lemma 2] specifies its one-schoenberg coefficients and hence the 3-Schoenberg coefficients can aso be cacuated. Pots (omitted here) of the corresponding Mercer coefficients for d {1, 3} show that DPPs with the kerne specified by the Askey, C 2 -Wendand, C 4 -Wendand, or spherica correation function are very far from the most repusive case. Tabe 1: Specia cases of Askey s truncated power function and C 2 -Wendand and C 4 - Wendand correation functions when d 3 and c = 1, where the two ast coumns specify the corresponding one-schoenberg coefficients. For a genera vaue of a scae parameter c > 0 (with c 2π in case of the Wendand functions), in the expressions for ψ, s shoud be repaced by s/c, whie in the expressions for β,1, shoud be repaced by c and the one-schoenberg coefficient shoud be mutipied by c. ψ β,1 ( = 1, 2,...) β 0,1 Askey (1 s) 3 + C 2 -Wendand (1 s) 4 +(4s + 1) C 4 -Wendand 1 3 (1 s)6 +(s(35s + 18) + 3) 6( 2 +2 cos() 2) π 4 1 4π 240( 2 + sin()+4 cos() 4) π 6 1 6π 8960( 4( 2 18)+3( 2 35) sin()+33 cos()) π π A fexibe spectra mode Suppose that the kerne of the DPP has Mercer coefficients λ,d = 1, = 0, 1,..., 1 + β exp ((/α) κ ) where α > 0, β > 0, and κ > 0 are parameters. Since a λ,d (0, 1), the DPP is we defined and has a density specified by (3.8). Since its kerne is positive definite, a finite subsets of S d are feasibe reaizations of the DPP. The mean number of points η may be evauated by numerica methods. The most repusive DPP is a imiting case: For any n {0, 1,...}, et η 0 = n =0 m,d, α = n, and β = 1/(nκ). Then, as κ, β exp ((/α) κ ) converges to 0 for n and to for > n. Thus λ,d 1 for n, λ,d 0 for > n, and η η 0. This imiting case corresponds to the case of X (η 0), the most repusive DPP consisting of η 0 points. Aso the homogeneous Poisson process is a imiting case: Note that for fixed d 1, the mutipicities m,d given by (4.4) satisfy the asymptotic estimate m,d (1 + ) d 1 as (again f 1 () f 2 () means that c d f 1 ()/f 2 () C d where 21

22 1.00 κ = 0.5 κ = 1 κ = Figure 3: Approximate pair correation functions for DPP modes with d = 2. Within each pane the fu ines from eft to right correspond to spectra modes with β = 100, 5, 1, 0.1, 0.01 and α chosen such that η 400 given the vaue of κ as indicated in the figure. The dotted ine corresponds to the most repusive DPP with η = 400. C d c d are positive constants). Hence, η = m,d λ,d =0 =0 (1 + ) d β exp((/α) κ ). Now, for a given vaue η = η 0 > 0, put κ = d and α = (η 0 β) 1/d. Then, for sufficienty arge β, (1 + ) d β exp((/α) d ) η 0. =0 Consequenty, as β, we obtain η η 0, whie λ,d 0, = 0, 1, 2... The mode covers a wide range of repusiveness as indicated in Figure 3. Compared to the mutiquadric mode in Figure 2 we immediatey notice that even with a moderate vaue of the exponent parameter (κ = 2) this mode aows us to come much coser to the most repusive DPP. The drawback of this mode is that neither the intensity nor the pair correation function is know anayticay. To approximate each pair correation function we have used (4.5) with β,d = λ,d m,d /η where η is evauated numericay Matérn covariance functions The Matérn correation function is of cass Ψ + and given by ( ψ(s) = 21 ν s ν ( s ) Kν, 0 s π, Γ(ν) c) c where ν (0, 1] and c > 0 are parameters and K 2 ν denotes the modified Besse function of the second kind of order ν, see [10, Section 4.5]. For ν = 1, ψ(s) = exp( s/c) 2 22

23 is the exponentia correation function. The nomencature Matérn function might be considered a bit ambitious here, since we are consideraby restricting the parameter space for ν. Indeed, it is true that any vaue of ν > 0 can be used when repacing the great circe distance with the chorda distance, but the use of this aternative metric is not contempated in the present work. Suppose R 0 = ψ is (the radia part of) the kerne for an isotropic DPP. Then the DPP becomes more and more repusive as the scae parameter c or the smoothness parameter ν increases, since g 0 then decreases. In the imit, as c tends to 0, g 0 tends to the pair correation function for a Poisson process. It can aso be verified that g 0 increases as ν decreases. For ν = 1, g 2 0(s) = 1 exp( 2s/c), so g 0(0) = 2/c > 0 and hence the DPP is ocay ess repusive than any other DPP with the same intensity and such that the sope for the tangent ine of its pair correation function at s = 0 is at most 2/c. For ν < 1, 2 g 0(0) =. That is, a singuarity shows up at zero in the derivative of R 0, which foows from the asymptotic expansions of K ν given in [1, Chapter 9]. Hence, the DPP is ocay ess repusive than any other DPP with the same intensity and with a finite sope for the tangent ine of its pair correation function at s = 0. Consequenty, the variance condition (4.21) is vioated for a ν (0, 1/2]. We are abe to derive a few more anaytica resuts: For d = 1 and ν = 1, (4.9) 2 yieds β 0,1 = c { ( 1 exp π )}, π c and Hence β,1 = 2 π { ( 1 + ( 1) +1 exp π )} c, = 1, 2,... c 1 + c 2 2 η max,1 = π c 1 1 exp ( π c which is a decreasing function of c, with range (1, ). Incidentay, [11] introduced what they ca a circuar Matérn covariance function and which for d = 1 has Mercer coefficients ), λ,1 = σ 2, = 0, 1,..., (α ν+1/2 ) where σ > 0, ν > 0, and α > 0 are parameters. Consider a DPP with the circuar Matérn covariance function as its kerne and with d = 1. This is we defined exacty when σ α ν+1/2. Then the mean number of points is η = = σ 2 (α 2 + ν+1/2, which is bounded by η max,1 = = 1 (1 + (/α) ν+1/2. 23

24 Figure 4: Pair correation functions for DPP modes with d = 1. Fu ines from eft to right correspond to circuar Matérn modes with ν = 0.5, 1, 2, 10, σ = α ν+1/2, and α = 31.8, 50, 75, chosen such that η = η max, The dotted ine corresponds to the most repusive DPP with η = 100. When ν is a haf-integer, the kerne and hence aso η and η max,1 are expressibe on cosed form, see [11]. Finay, considering the case σ = α ν+1/2 (equivaenty η = η max,1 ), then 1 λ,1 =, = 0, 1,..., (1 + (/α) 2 ν+1/2 ) and we see that the DPP never reaches the most repusive DPP except in the imit where η max,1 1. Figure 4 shows four different pair correation functions corresponding to different vaues of ν for the circuar Matérn mode with η = η max,1 (i.e. σ = α ν+1/. For vaues ν > 10 the curves become amost indistinguishabe from the one with ν = 10, so we have omitted these from the figure. The eft most curve (ν = 1/2) is in fact amost numericay identica to the ordinary Matérn mode introduced above with ν = 1/2 and η max,1 = 100. Since ν = 1/2 corresponds to the most repusive ordinary Matérn mode we see that the circuar Matérn mode covers a much arger degree of repusiveness than the ordinary Matérn mode. 5 Concuding remarks In this paper, we have considered determinanta point processes (DPPs) on the d-dimensiona unit sphere S d. We have shown that DPPs on spheres share many properties with DPPs on R d, and these properties are simper to estabish and to expoit for statistica purposes on S d due to compactness of the space. For DPPs with a distribution specified by a given kerne (a compex covariance function of finite trace cass and which is square integrabe), we have deveoped a suitabe Mercer (spectra) representation of the kerne. In particuar, we have studied the case of DPPs with a distribution specified by a continuous isotropic kerne. Such kernes can be expressed by a Schoenberg representation in terms of countabe inear combination of Gegenbauer poynomias, and a precise connection between 24

25 the Schoenberg representation and the Mercer representation has been presented. Furthermore, the trade-off between the degree of repusiveness and the expected number of points in the mode has been estabished, and the most repusive isotropic DPPs have been identified. We have used the connection between the Schoenberg and Mercer representations to construct a number of tractabe and fexibe parametric modes for isotropic DPPs. We have considered two different modeing approaches where we either work with a cosed form expression for the correation function ψ or with its Mercer/spectra representation. With the former approach the mutiquadric mode seems to be the most promising mode with some fexibiity, and the cosed form expression for ψ opens up for computationay fast moment based parameter estimation in future work. The two main drawbacks is that this cass cannot cover the most extreme cases of repusion between points and simuation requires truncation of a (possiby) infinite series, which in some cases may be probematic. However, in our experience the truncation works we in the most interesting cases when we are not too cose to a Poisson point process. In that case the Schoenberg coefficients decay very sowy and it becomes computationay infeasibe to accuratey approximate the d-schoenberg coefficients. The fexibe spectra mode we have deveoped overcomes these two drawbacks. It covers the entire range of repusiveness from the ack of repusion in the Poisson case to the most repusive DPP, and it is straightforward to generate simuated reaizations from this mode. The main drawback here is that the intensity and pair correation function ony can be evauated numericay making moment based inference more difficut, and the parameters of the mode may be harder to interpret. We defer for another paper how to construct anisotropic modes and to perform statistica inference for spatia point pattern datasets on the sphere. In brief, smooth transformations and independent thinnings of DPPs resuts in new DPPs, whereby anisotropic DPPs can be constructed from isotropic DPPs. For d = 2 particuar forms for anisotropy shoud aso be investigated such as axia symmetry (see e.g. [16, 13]) meaning that the kerne is invariant to shifts in the poar ongitude. We have deveoped software in the R anguage [23] to hande DPPs on the sphere as an extension to the spatstat package [2] and it wi be reeased in a future version of spatstat. Unti officia reease in spatstat the code can be obtained by sending an emai to the authors. Some figures have been created with the R package ggpot2 [29]. Acknowedgments Supported by the Danish Counci for Independent Research Natura Sciences, grant , Mathematica and Statistica Anaysis of Spatia Data, by Proyecto Fondecyt Reguar from the Chiean Ministry of Education, and by the Centre for Stochastic Geometry and Advanced Bioimaging, funded by a grant (8721) from the Vium Foundation. 25